Introduction
Metamaterials, which are engineered composite media, can be designed and constructed to exhibit unconventional characteristics and new response functions for the constitutive parameters. In the last decade, there has been growing interest in the study of the metamaterials in the research community both theoretically and experimentally due to their exciting potential applications ranging from perfect lenses [1], cloaking devices [2-4] and slow lights [5] to sub-wavelength optical waveguides [6,7], ultrasensitive sensors [8-10], microstructured magnetic materials [11,12] and circular dichroism [13-16]. Recently, Gansel et al. [17-19] have succeeded in developing a broadband circular polarizer using gold single-helical metamaterials; Wu et al. studied metallic helix arrays theoretically and experimentally in microwave ranges [20,21].
Circular polarization of light is attractive for applications in reflective color displays [22-24], life science microscopy, and photography [25,26]. Generally, there are two ways of obtaining circular polarized light: one is using a linear polarizer and a quarter-wave plate [27], which is the most common method in optics; the other is utilizing cholesteric liquid crystals (CLC) [28-31], which are self-assembled photonic crystals formed by rod-like molecules. Compared with these two methods, the helical metamaterials has advantages of broad wavelength ranges and compact structures, which are convenient to be integrated with other optical devices.
As far as we know, however, most of the researches on the helical metamaterials were under the cases of single-, circular-helixes, and normal incidences. In this paper, recent simulation works by the finite difference time domain (FDTD) method on the helical metamaterials are reviewed, which mainly included the optical performances of double-, three-, four-helical metamaterials, performances of the elliptical-helical metamaterials, and the polarization properties under the condition of oblique incidences [32-36]. The results demonstrate that the double-helical metamaterials has operation bands more than 50% broader than the single-helical structures. But both of them have low signal-to-noise ratios about 10 dB. The three- and four-helical metamaterials have a significant improvement in overall performance. For elliptical-helixes, simulation results showed that the output light can have elliptical polarization states. On the condition of oblique incidences, a novel property of tunable polarization states occurred in the helical metamaterials, which could have much broader potential applications such as tunable optical polarizers, tunable beam splitters, and tunable optical attenuators.
Simulation models
Figure 1 shows the schematic diagrams of a single-helical metamaterials. It consists of left-handed helical nanowire arrays and a substrate of silica. In Fig. 1, DW, NH, SG, LH, and DH stand for the diameter of the wire, the number of the helix-periods, the spacing of the grid, the length of the helix-period, and the diameter of the helix, respectively. Two kinds of polarized lights, left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) lights are used as the excitation sources to irradiate the metamaterials along the Z direction, respectively. To simplify the simulation, a broadband Gaussian-modulated pulsed light source is used as the excitation source. The perfectly matched layers (PMLs) [37] were used as the boundary conditions. The boundaries along X and Y directions were confined with the periodic boundary conditions [38], due to the periodicity of the nanowire metamaterials. During the calculation, the dielectric function of the metals was described by the Lorentz-Drude model [39].
Simulation results and analyses
Double-helical metamaterials
Single- and double-helical metamaterials with two different metals, gold (Au) and aluminum (Al) were simulated using the FDTD method. Figure 2 shows the schematic diagram of a double-helical metamaterials. The parameters of the metamaterial structure are: DW= 50 nm, NH= 3, SG= 190 nm, LH= 200 nm, and DH= 100 nm, respectively. Their optical performances are shown in Fig. 3. Figures 3(a) and 3(c) are transmittances of the circular polarizers for the LCP and RCP light beams and the extinction ratio as functions of the wavelength for the single-helical structure of Au and Al nanowires, respectively. Figures 3(b) and 3(d) are the results for the double-helical structure of Au and Al nanowires, respectively.
The operation regions of the circular polarizers are defined as the wavelength regions, in which the extinction ratio is not less than 1/e of its peak value. In the operation regions, the average transmittances for the RCP light and the average extinction ratios with respect to single- and double- helix metamaterials and different metals are shown in Fig. 3 and listed in Table 1. It is very clear that the operation regions of the circular polarizers with the double-helical structures are more than 50% broader than those of the single-helical structures.
The mechanism may be come from the coupling effect in metamaterials. In a general model of metamaterial, one single-helix can be seen as an artificial atom. The double-helix structure can be seen as molecule composed from two coupled helix atoms. For two coupled helix resonators, the mode can be seen as hybridized mode from single-helix resonator. Hybridization usually results broadening of spectrum of resonance mode. Then it may give an explanation why the double-helix has much broader band than the single-helix [40,41].
Tab.1 Comparison of optical performances between single- and double-helical metamaterials |
| operation regions/μm | average transmittances of RCP light/% | average extinction ratios | |||
|---|---|---|---|---|---|
| Au | single-helix | 0.72-1.00 | 63 | 36∶1 | |
| double-helix | 0.75-1.30 | 65 | 25∶1 | ||
| Al | single-helix | 0.42-0.79 | 63 | 39∶1 | |
| double-helix | 0.44-1.10 | 56 | 31∶1 | ||
Three- and four-helical metamaterials
The single-, double-, three-, and four-helical Al metamaterials were simulated using the FDTD method. Figures 4(a) and 4(b) show the schematic diagrams of the three- and four-helix, respectively. The optical performances of the incident and transmitted lightwaves are shown in Fig. 5. Figure 5(a) shows schematic diagram of the amplitude (Amp.) and the phase angle (θ) of the electric vector; Fig. 5(b) shows the amplitude and the phase angle for the LCP incident light; Figs. 5(c), 5(e), 5(g), and 5(i) are the results for the transmitted lights through the single-, double-, three-, and four-helical structures, respectively; Figs. 5(d), 5(f), 5(h), and 5(j) are the S/N ratios for them, respectively. From the simulation results, it is very clear that the S/N ratios of the three- and four-helixes are two orders higher than those of single- and double-helical ones.
Elliptical-helical metamaterials
Figure 6 shows the schematic diagram of the elliptically single-helical metamaterials, in which LEH, SHE, SGL, and SGS stand for long-axis of elliptical helix, short-axis of elliptical helix, and spacing of grids in long-axis direction and in short-axis direction, respectively. The axial ratio (the length of the major semiaxis to that of the minor semiaxis) is 2∶1. The metal is Al. The excitation sources are orthogonal left-elliptically polarized (LEP) light and right-elliptically polarized (REP) light propagating along Z axis in free space. They are represented with the following Jones vectors [20]:
The parameters’ values are: DW= 30 nm, NH= 3, LH= 200 nm, LEH= 80 nm, SEH= 40 nm, SGL= 310 nm, and SGS= 170 nm. Figure 6(a) shows the transmittance and the extinction ratio of the structure, in which the operation region is 680-1100 nm. In the region, the average transmittance of LEP light is 74% and the average extinction ratio is 11.1 dB. It is obvious that this metamaterials has a giant elliptical dichroism. To analyze the polarization states of the transmitted LEP light in detail, the Poincaré sphere [42] is plotted in Fig. 6(b), in which these red points refer to different wavelengths. The coordinates of the points on the Poincaré sphere, their corresponding polarization states and the conversions of LEP are listed in Table 2. From the calculation results, it is clear that the axial ratios of the transmitted LEP light are all around 2∶1, which is the same as that of the incident LEP light.
Tab.2 Polarization states of transmitted LEP light |
| WL/µm | coordinates | AR | conversion of LEP |
|---|---|---|---|
| 1.07 | (-0.58, 0.36,-0.72) | 1.95 | 4.1% |
| 0.93 | (-0.63, 0.31,-0.70) | 2.12 | 3.6% |
| 0.83 | (-0.62, 0.33,-0.70) | 2.09 | 4.4% |
| 0.75 | (-0.58, 0.31,-0.74) | 1.95 | 3.6% |
| WL: Wavelength, AR: axial ratio | |||
Performances with oblique incidences
The helical metamaterials with different incident angles were simulated using the FDTD method. The parameters and the simulation results including the axial ratios, the extinction ratio, and the transmittances of LCP light are summarized in Table 3 and shown in Fig. 7. Figures 7(a)-7(e) are transmittances of the helical metamaterials with different incident angles 20°, 10°, 0°, -10°, and -20°, respectively. In Fig. 7(f), the Poincaré sphere was used to analyze the polarization states of the transmitted lights at wavelength 0.68 μm. Different color points refer to different incident angles. The closer the points are to the pole, the closer the axial ratios are to the value of 1∶1. From these simulation results, it is clear that the transmitted light has tunable polarization states with changing the incident angles.
Tab.3 Parameters and simulation results with different incident angles |
| angle of incidence/(°) | axial ratio | transmittances of LCP light/% | extinction ratio | diagrams |
|---|---|---|---|---|
| 20 | 1∶0.93 | 57 | 5.04∶1 | Fig.7(a), “<InlineMediaObject OutputMedium="Online"><ImageObject FileRef="images\hcm0000446669.jpg" ScaleToFitWidth="10cm" ScaleToFit="1"/></InlineMediaObject><InlineMediaObject OutputMedium="All"><ImageObject FileRef="images\hcm0000446668.tif" ScaleToFit="1" ScaleToFitWidth="10cm"/></InlineMediaObject>” in Fig. 7(f) |
| 10 | 1∶0.80 | 65 | 6.78∶1 | Fig.7(b), “<InlineMediaObject OutputMedium="Online"><ImageObject FileRef="images\hcm0000446667.jpg" ScaleToFitWidth="10cm" ScaleToFit="1"/></InlineMediaObject><InlineMediaObject OutputMedium="All"><ImageObject FileRef="images\hcm0000446666.tif" ScaleToFit="1" ScaleToFitWidth="10cm"/></InlineMediaObject>” in Fig. 7(f) |
| 0 | 1∶0.67 | 69 | 7.58∶1 | Fig. 7(c), “<InlineMediaObject OutputMedium="Online"><ImageObject FileRef="images\hcm0000446665.jpg" ScaleToFitWidth="10cm" ScaleToFit="1"/></InlineMediaObject><InlineMediaObject OutputMedium="All"><ImageObject FileRef="images\hcm0000446664.tif" ScaleToFit="1" ScaleToFitWidth="10cm"/></InlineMediaObject>” in Fig. 7(f) |
| -10 | 1∶0.66 | 68 | 7.71∶1 | Fig. 7(d), “<InlineMediaObject OutputMedium="Online"><ImageObject FileRef="images\hcm0000446663.jpg" ScaleToFitWidth="10cm" ScaleToFit="1"/></InlineMediaObject><InlineMediaObject OutputMedium="All"><ImageObject FileRef="images\hcm0000446662.tif" ScaleToFit="1" ScaleToFitWidth="10cm"/></InlineMediaObject>” in Fig. 7(f) |
| -20 | 1∶0.59 | 62 | 6.22∶1 | Fig. 7(e), “<InlineMediaObject OutputMedium="Online"><ImageObject FileRef="images\hcm0000446661.jpg" ScaleToFitWidth="10cm" ScaleToFit="1"/></InlineMediaObject><InlineMediaObject OutputMedium="All"><ImageObject FileRef="images\hcm0000446680.tif" ScaleToFit="1" ScaleToFitWidth="10cm"/></InlineMediaObject>” in Fig. 7(f) |
| DW= 30 m, NH= 3, SG= 200 nm, LH= 200 nm, DH= 100 nm. | ||||
Fig.7 Optical performances of the helical metamaterials with different incident angles. (a)-(e) are for the incident angles of 20°, 10°, 0°, -10°, and -20°, respectively; (f) is comparison of the transmitted light’s polarization states (represented on the Poincaré sphere). The blue, green, red, cyan, and pink points refer to the incident angles of 20°, 10°, 0°, -10°, and -20°, respectively |
Conclusions
In summary, this paper reviewed recent simulation works in the helical metamaterials, which mainly included the optical performances of double-, three-, four-helical metamaterials, performances of the elliptical-helical metamaterials, and the polarization properties under the condition of oblique incidences. The results demonstrate that the double-helical metamaterials has operation bands more than 50% broader than the single-helical structures. But both of them have low signal-to-noise ratios about 10 dB. The three- and four-helical metamaterials have a significant improvement in overall performance. For elliptical-helixes, simulation results show that the output light can have elliptical polarization states. On the condition of oblique incidences, a novel property of tunable polarization states occurred in the helical metamaterials, which could have much broader potential applications such as tunable optical polarizers, tunable beam splitters, and tunable optical attenuators.
It is certain that there are still some challenges on the fabrication of the real 3D helical metamaterials. In our opinions, two aspects of the helical metamaterials will become the development directions. One is improvements of nanofabrication techniques; the other is design of simple 2D chiral metamaterials, which also has circular dichroism, but need not complex fabrication processes. In a word, with developments of the metamaterials, more and more novel structures and devices could be realized in the near future.